Bipartite graphs model relationships between two different sets of entities, like actor-movie, user-item, and author-paper. The butterfly, a 4-vertices 4-edges $2\times 2$ bi-clique, is the simplest cohesive motif in a bipartite graph and is the fundamental component of higher-order substructures. Counting and enumerating the butterflies offer significant benefits across various applications, including fraud detection, graph embedding, and community search. While the corresponding motif, the triangle, in the unipartite graphs has been widely studied in both static and temporal settings, the extension of butterfly to temporal bipartite graphs remains unexplored. In this paper, we investigate the temporal butterfly counting and enumeration problem: count and enumerate the butterflies whose edges establish following a certain order within a given duration. Towards efficient computation, we devise a non-trivial baseline rooted in the state-of-the-art butterfly counting algorithm on static graphs, further, explore the intrinsic property of the temporal butterfly, and optimize the process with a compact data structure and smart pruning strategies. The time complexity is proved to be significantly reduced without compromising on space efficiency. In addition, we generalize our algorithms to practical streaming settings and multi-core computing architectures. Our extensive experiments on 11 large-scale real-world datasets demonstrate the efficiency and scalability of our solutions.

翻译：二分图建模两类不同实体集（如演员-电影、用户-商品、作者-论文）之间的关联关系。蝶形结构（butterfly），即4个顶点和4条边构成的$2\times 2$双团，是二分图中最简凝聚性子图基序，也是高阶子结构的基础构成单元。蝶形结构的计数与枚举在欺诈检测、图嵌入和社区搜索等众多应用中具有重要价值。尽管对应单分图中的三角形基序已在静态和时序场景下得到广泛研究，但蝶形结构向时序二分图的扩展仍属空白。本文研究时序二分图中的蝶形计数与枚举问题：统计并枚举各边在给定时间窗口内按特定顺序建立的蝶形结构。为实现高效计算，我们基于静态图最先进蝶形计数算法设计了非平凡基线方法，进一步探究时序蝶形的内在特性，通过紧凑数据结构和智能剪枝策略优化计算流程。理论分析表明，该方法在保持空间效率的同时显著降低了时间复杂度。此外，我们将算法推广至流式处理场景与多核计算架构。在11个大规模真实数据集上的广泛实验验证了所提方案的效率与可扩展性。